Answer
a. $\sim p\wedge q$
b.
$\begin{array}{lllll}
p & q & \sim p & \sim p\wedge q & \\
\hline T & T & F & F & \\
T & F & F & F & \\
F & T & T & T & \\
F & F & T & F &
\end{array}$
c. when p is false, and q is true.
Work Step by Step
Restructuring the statement:
NOT( you did the dishes) AND (you left the room in a mess)
a.
p: You did the dishes;
q: You left the room in mess
$\sim p\wedge q$
b.
Set up a truth table for two inputs, p and q:
$\begin{array}{llll}
p & q & ... & ...\\
\hline T & T & & \\
T & F & & \\
F & T & & \\
F & F & &
\end{array}$
Next column: $\sim p$,
negation of column 1,
$\begin{array}{lllll}
p & q & \sim p & & \\
\hline T & T & F & & \\
T & F & F & & \\
F & T & T & & \\
F & F & T & &
\end{array}$
Next column: $\sim p\wedge q$,
conjunction of columns 3 and 2
$\begin{array}{lllll}
p & q & \sim p & \sim p\wedge q & \\
\hline T & T & F & F & \\
T & F & F & F & \\
F & T & T & T & \\
F & F & T & F &
\end{array}$
c.
The statement is true (see row 3)
when p is false, and q is true.
